Fix \(\alpha ,\beta ,\alpha ^{\prime },\beta ^{\prime }\in \mathbb {N}I\)  such that \(\alpha +\beta =\alpha ^{\prime }+\beta ^{\prime }\) .
Let \(\mathcal {N}\)  be the set of \((\alpha _1,\alpha _2,\beta _1,\beta _2)\in (\mathbb {N}I)^4\)  such that \(\alpha _1+\alpha _2=\alpha , \beta _1+\beta _2=\beta , \alpha _1+\beta _1=\alpha ^{\prime }, \alpha _2+\beta _2=\beta ^{\prime }\) .
For any \(A\in \mathcal {Q}_{{\alpha }}\)  and \(B\in \mathcal {Q}_{{\beta }}\) , Lusztig ([1]}) proved the compatibility of induction functors and restriction functors:
\(&&\mathrm {Res}^{\alpha ^{\prime }+\beta ^{\prime }}_{\alpha ^{\prime },\beta ^{\prime }}\mathrm {Ind}^{\alpha +\beta }_{\alpha ,\beta }(A\boxtimes B)\\&\simeq &\bigoplus _{\lambda \in \mathcal {N}}(\mathrm {Ind}^{\alpha ^{\prime }}_{\alpha _1,\beta _1}\times \mathrm {Ind}^{\beta ^{\prime }}_{\alpha _2,\beta _2})(\tau _{\lambda })_{!}(\mathrm {Res}^{\alpha }_{\alpha _1,\alpha _2}(A)\times \mathrm {Res}^{\beta }_{\beta _1,\beta _2}(B)) [-(\alpha _2,\beta _1)](-\frac{(\alpha _2,\beta _1)}{2})\nonumber .\) 
